Finite Group Databasebeta

← #28345717

Subgroups — #28345717

Order 13696.

754 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic
2C2×#16812140C211yesmaximal, characteristic, complemented
3C2×#16812085C211yesmaximal, characteristic, complemented
4#20292788C212yesmaximal, complemented
5#20292786C212yesmaximal, complemented
6#20292781C212yesmaximal, complemented
7#20292776C212yesmaximal, complemented
8#20292629C211yesmaximal, characteristic, complemented
9#20292628C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C2×#13133817—21no—
16C2×#13133817C2211yescharacteristic
17#41×D107—24no—
18#41×D107—24no—
19C2×Q8×D107C2211yescharacteristic
20C2×D4×D107—21no—
21C2×D4×D107C2211yescharacteristic
22C22×C4×D107—21no—
23C22×C4×D107C2211yescharacteristic
24#16812140—24no—
25C2×#13133808C2212yescomplemented
26C2×#13133805C2212yescomplemented
27C2×#13133802C2211yescharacteristic, complemented
28#16812085—21no—
29#16812085—21no—
30#16812085C2212yes—
31#16812085C2212yescomplemented
32C2×C4⋊Dic107C2211yescharacteristic, complemented
33C2×Dic107⋊C4—21no—
34C2×(C2×D107)⋊C4—21no—
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812031C2212yescomplemented
38#16812030C2212yescomplemented
39(C2×D107)⋊C8C2212yes—
40(C2×D107)⋊C8C2212yescomplemented
41C2×#41×C107C2211yescharacteristic
42C22×Q8×C107C2211yescharacteristic, complemented
43#137×C107—24no—
44C2×Q16×C107C2212yescomplemented
45C2×SD16×C107C2212yescomplemented
46C2×M16×C107C2211yescharacteristic, complemented
47C23×D107—21no—
48C23×D107D411yescharacteristic
49#13133819—22no—
50#13133819—22no—
51#13133818—28no—
52#13133818—22no—
53#13133818D414yes—
54#13133817—28no—
55#13133817—22no—
56#13133817—22no—
57C2×Dic214—21no—
58C2×Dic214C2311yescharacteristic
59Q8×D107—22no—
60Q8×D107—22no—
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66D4×D107—28no—
67D4×D107—22no—
68D4×D107—22no—
69C22×Dic107—21no—
70C22×Dic107—21no—
71C22×Dic107D411yescharacteristic
72C2×C4×D107—28no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—21no—
78C2×C4×D107D411yescharacteristic
79C2×C4×D107D411yescharacteristic
80C2×C4×D107C2311yescharacteristic
81C2×C4×D107C2311yescharacteristic
82#13133808—24no—
83#13133805—24no—
84#13133802—22no—
85C2×C107⋊C8C2312yes—
86C4⋊Dic107—21no—
87C4⋊Dic107C2312yes—
88Dic107⋊C4—22no—
89Dic107⋊C4—22no—
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—22no—
92#41×C107—22no—
93#41×C107D414yes—
94C2×Q8×C107—21no—
95C2×Q8×C107—22no—
96C2×Q8×C107—22no—
97C2×Q8×C107C2312yes—
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—21no—
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107—21no—
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104Q16×C107—24no—
105SD16×C107—24no—
106M16×C107—22no—
107C2×C8×C107C2312yes—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—24no—
111C22×D107—22no—
112C22×D107—21no—
113C22×D107—22no—
114C22×D107—41no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107C2×D411yescharacteristic
118C22×D107C2×D411yescharacteristic
119Q8×C107—21no—
120Q8×C107—22no—
121Q8×C107—22no—
122Q8×C107—42no—
123Q8×C107C2×D412yes—
124D4×C107—24no—
125D4×C107—21no—
126D4×C107C2×D412yes—
127C2×C4×C107—21no—
128C2×C4×C107—24no—
129C2×C4×C107—22no—
130C2×C4×C107—42no—
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D412yes—
133C2×C4×C107C2×D412yes—
134C2×C4×C107C2411yescharacteristic
135C22⋊D107—24no—
136C22⋊D107—24no—
137C22⋊D107—24no—
138C2×Dic107—24no—
139C2×Dic107—22no—
140C2×Dic107—21no—
141C2×Dic107—22no—
142C2×Dic107—42no—
143C2×Dic107C2×D411yescharacteristic
144C2×Dic107C2×D411yescharacteristic
145D428—24no—
146D428—21no—
147D428C2×D412yes—
148C4×D107—24no—
149C4×D107—24no—
150C4×D107—22no—
151C4×D107—24no—
152C4×D107—22no—
153C4×D107—21no—
154C4×D107—21no—
155C4×D107—21no—
156C4×D107—21no—
157Dic214—24no—
158Dic214—21no—
159Dic214C2×D412yes—
160C8×C107—22no—
161C107⋊C8—22no—
162C22×C107—21no—
163C22×C107—21no—
164C22×C107—22no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107#12012yes—
168C22×C107C22×D411yescharacteristic
169C2×D107—21no—
170C2×D107—21no—
171C2×D107—21no—
172C2×D107—22no—
173C2×D107—21no—
174C2×D107—21no—
175C2×D107—44no—
176C2×D107—48no—
177C2×D107—41no—
178C2×D107—41no—
179C2×D107#13712yescomplemented
180C4×C107—22no—
181C4×C107—42no—
182C4×C107#12012yes—
183C4×C107C22×D411yescharacteristic
184C4×C107C22×D411yescharacteristic
185Dic107—21no—
186Dic107—22no—
187Dic107—21no—
188Dic107—42no—
189C2×C107—22no—
190C2×C107—42no—
191C2×C107C2×#12011yescharacteristic
192C2×C107C2×#13711yescharacteristic
193C2×C107#57511yescharacteristic
194D107—22no—
195D107—41no—
196D107—42no—
197#3007—1071nomaximal, Sylow
198C107#300711yescharacteristic, complemented, Sylow
199C22×#41—1071no—
200C2×#137—1071no—
201C2×#137D10711yescharacteristic, complemented
202C2×C22⋊Q8—1071no—
203#452—1072no—
204#452—1072no—
205#451—1072no—
206#450—1072no—
207#417—1071no—
208#417—1071no—
209#407—1071no—
210C2×#41—1071no—
211C2×#41—1071no—
212C2×#41—1071no—
213C2×#41—2141no—
214C2×#41—2141no—
215C2×#41—2144no—
216C2×#41—2144no—
217C2×#41C2×D10711yescharacteristic
218C22×Q8—1071no—
219C22×Q8C2×D10711yescharacteristic, complemented
220C22×D4—1071no—
221C22×D4—2141no—
222C23×C4—1071no—
223C23×C4—2141no—
224#137—24no—
225#137—2144no—
226C2×Q16—1072no—
227C2×Q16C2×D10712yescomplemented
228C2×SD16—1072no—
229C2×SD16C2×D10712yescomplemented
230C2×M16—1071no—
231C2×M16C2×D10711yescharacteristic, complemented
232C22⋊Q8—1072no—
233C22⋊Q8—1072no—
234C22⋊Q8—2141no—
235C22⋊Q8—2141no—
236C2×C4⋊C4—1071no—
237C2×C4⋊C4—2141no—
238C2×C22⋊C4—2141no—
239#103—1072no—
240#103—1072no—
241#102—1072no—
242#102—1072no—
243C22⋊C8—1072no—
244C22⋊C8—1072no—
245C24—1071no—
246C24—2141no—
247#41—22no—
248#41—1074no—
249#41—2148no—
250#41—2142no—
251#41—2148no—
252#41—2142no—
253#41—2142no—
254#41—2142no—
255#41—2142no—
256#41C22⋊D10714yes—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—1071no—
261C2×Q8—2141no—
262C2×Q8—2142no—
263C2×Q8—2142no—
264C2×Q8C22×D10712yes—
265C2×Q8C22×D10711yescharacteristic
266C2×D4—21no—
267C2×D4—1071no—
268C2×D4—2141no—
269C2×D4—2141no—
270C2×D4—2142no—
271C2×D4—2148no—
272C2×D4—2141no—
273C2×D4—2141no—
274C2×D4—2142no—
275C2×D4C22×D10711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1071no—
279C22×C4—1071no—
280C22×C4—1071no—
281C22×C4—1071no—
282C22×C4—1071no—
283C22×C4—2141no—
284C22×C4—2141no—
285C22×C4—2148no—
286C22×C4—2141no—
287C22×C4—2141no—
288C22×C4—2142no—
289C22×C4—2141no—
290C22×C4—2142no—
291C22×C4C22×D10711yescharacteristic
292Q16—24no—
293Q16—2144no—
294SD16—24no—
295SD16—2144no—
296M16—22no—
297M16—2142no—
298C2×C8—1072no—
299C2×C8C22×D10712yes—
300C4⋊C4—1072no—
301C4⋊C4—2142no—
302C4⋊C4—2142no—
303C4⋊C4—2141no—
304C22⋊C4—2142no—
305C22⋊C4—2142no—
306C23—21no—
307C23—1071no—
308C23—1071no—
309C23—4281no—
310C23—4282no—
311C23—4282no—
312C23—2142no—
313C23—2144no—
314C23—2142no—
315C23—2141no—
316C23D4×D10711yescharacteristic
317Q8—21no—
318Q8—22no—
319Q8—22no—
320Q8—42no—
321Q8—1072no—
322Q8—2144no—
323Q8—2141no—
324Q8C2×C22⋊D10712yes—
325D4—24no—
326D4—21no—
327D4—1072no—
328D4—2144no—
329D4—2144no—
330D4—2144no—
331D4—2144no—
332D4—2141no—
333D4C2×C22⋊D10712yes—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—1071no—
339C2×C4—1071no—
340C2×C4—4282no—
341C2×C4—2144no—
342C2×C4—2144no—
343C2×C4—2142no—
344C2×C4—2144no—
345C2×C4—2142no—
346C2×C4—2141no—
347C2×C4—2142no—
348C2×C4—2142no—
349C2×C4—2144no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4D4×D10711yescharacteristic
355C2×C4D4×D10712yes—
356C2×C4C2×C22⋊D10712yes—
357C2×C4C23×D10711yescharacteristic
358C8—22no—
359C8—2142no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—1072no—
366C22—4284no—
367C22—4288no—
368C22—4281no—
369C22—4281no—
370C22—2141no—
371C22—2141no—
372C22—2142no—
373C22—2141no—
374C22—2141no—
375C22—2141no—
376C22#1681207712yes—
377C22C2×D4×D10711yescharacteristic
378C4—22no—
379C4—42no—
380C4—4282no—
381C4—2141no—
382C4—2141no—
383C4—2142no—
384C4#1681207712yes—
385C4C2×D4×D10711yescharacteristic
386C4C22×C22⋊D10711yescharacteristic
387C2—22no—
388C2—42no—
389C2—4282no—
390C2—4281no—
391C2—2142no—
392C2C2×#1681207711yescharacteristic
393C2#137×D10711yescharacteristic
394C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×D107C2
hypercenterC2×#137D107
hypocenterC107#3007
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. C2×#16812085C2semidirect
4. #20292788C2semidirect
5. #20292786C2semidirect
6. #20292781C2semidirect
7. #20292776C2semidirect
8. #20292629C2semidirect
9. #20292628C2semidirect
10. #20292590C2semidirect
11. C2×#137×C107C2semidirect
25. C2×#13133808C22semidirect
26. C2×#13133805C22semidirect
27. C2×#13133802C22semidirect
31. #16812085C22semidirect
32. C2×C4⋊Dic107C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812031C22semidirect
38. #16812030C22semidirect
40. (C2×D107)⋊C8C22semidirect
42. C22×Q8×C107C22semidirect
44. C2×Q16×C107C22semidirect
45. C2×SD16×C107C22semidirect
46. C2×M16×C107C22semidirect
179. C2×D107#137semidirect
198. C107#3007semidirect
201. C2×#137D107semidirect
219. C22×Q8C2×D107semidirect
227. C2×Q16C2×D107semidirect
229. C2×SD16C2×D107semidirect
231. C2×M16C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#41×D1072
C2×#168121402
C2×#168120852
#202927882
#202927862
#202927812
#202927762
#202926292
#202926282
#202925902
C2×#137×C1072
#3007107